What is a Variable?
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Basics on the topic What is a Variable?
A variable is a symbol that is used to hold a value or values in equations and inequalities. It is usually represented by a letter like x, y, or any letter which best represents the unknown value or values. Variables are used in writing algebraic expressions, together with constants, operators, and coefficients, to solve real-world or mathematical problems. Since variables represent unknown numerical values, we can perform operations on them just like we do with numbers. This means that the operations addition, subtraction, multiplication, and division apply to variables as well. Learn what a variable is by helping the two Junior Explorers, Jessica and Cello, earn the prestigious Blue Diamond Survival Patch on their camping trip in the New Mexico desert. Common Core Reference: CCSS.MATH.CONTENT.6.EE.B.6
Transcript What is a Variable?
Mathematically variables in equations
Scout master Blanco and two Junior Explorers are camping in the New Mexico desert. The two Junior Explorers, Jessica and Cello, must successfully survive a night in the desert alone in order to earn the most prestigious badge: The Blue Diamond Survival Patch. They’re accompanied by their trusty handbook, which will advise them on how to deal with the desert’s perilous variables. As night falls, the Junior Explorers are left with their handbook, their courage and their wits. Mr. Blanco is planning the bonfire for the next night. He of course, references his survival handbook.
It says that, in order to make an awesome bonfire. The fire needs 5 pieces of wood to start, and each additional piece adds an hour of burning time. It also says that the number of pieces of wood for an ideal campfire is 11. How can we express this mathematically? To help Mr. Blanco build the ideal campfire, we can substitute other information we know into our equation. There's one piece of information missing, which we call a variable. But this would be annoying to write, so we can just use a letter for the variable, i.e. "x", but make sure to remember what it stands for. After deciding how much wood he’ll need for the next night’s bonfire, Mr. Blanco goes off into the night in search of wood.
Second example - Steaks for coyoties
Now that the Explorers are all by their lonesome, the trepidation sets in. What's this? Is that a coyote? Cello's in luck? To help him in his quest for the Blue Diamond Survival Patch, Cello can use his handbook and the steaks his mom packed for the trip. Cello looks in his handbook and it says that coyotes need one steak to keep them occupied long enough so you can escape. The handbook suggests to set up an equation with the help of variables. The total number of steaks minus the number of coyotes you can distract equals the steaks available for the campers to eat. We know the total number of steaks, 15 the campers and Mr. Blanco only need 5 steaks. The total number of distracted coyotes is our final unknown, which we'll call 'c', for coyotes. Now that we're down to our final variable, Cello can now solve his coyote problem.
Third example - Rattlesnakes
Meanwhile, while preparing for bed, Jessica is interrupted by a familiar sound coming from outside her tent...it must be a rattlesnake...or two...or….MORE?! The only thing Jessica is sure about is that there are rattlesnakes outside her tent, and she brought 4 sacks for catching snakes. She quickly consults her Junior Explorer handbook and it tells her that the best way to deal with rattlesnakes is with a snake scoop and a burlap sack. She can put up to three snakes in a sack. How many snakes could Jessica bag with the sacks she’s brought? We take the total number of sacks Jessica brought, 4 and multiply by the number of snakes she can put in each sack, 3. Finally, we can name our variable, the number of snakes Jessica can catch, 'S'.
In the morning, and without incident, the two Junior Explorers meet at the center of the camp. When they notice a trail. How to make coyote noises? How to sound like a rattlesnake? The Junior Explorers follow the trail of items and it leads them to Mr. Blanco?!? That trickster! These Junior Explorers really deserve their Blue Diamond Survival Patches!
What is a Variable? exercise
Explain what a variable is.Hints
For example, Paul is $12$ years old. The age of Paul's sister, Anne, is unknown. Together they are $21$ years old.
This leads to the equation $12+a=21$, where $a$ is the variable representing Anne's age.
Say Paul has two apples and five pears and both of them have the same price, $x$. If he pays $1.40$ dollars in total, then the corresponding equation is $2x+5x=1.4$.
Because both summands have the factor $x$ in common, we can factor it to get $2x+5x=(2+5)x=7x$.
An example of an algebraic equation is $2x-2=12$.Solution
A variable is a symbol, often $x$, $y$, or any letter used to represent an unknown value. For instance,
- $c$ for coyote
- $s$ for snake
Variables can be used in algebraic expressions, together with constants, operators, etc. Using variables can be useful when solving word problems.
Variable can be treated in the same way as numbers, as they represent quantities:
- We can add or subtract them $2x+3x=5x$ or $5s-3s=2s$.
- We can multiply them $(2x)(3x)=6x^2$.
- We can also divide them $(6x^2)\div(3x)=2x$.
Find the right equationHints
Not every variable must be $x$ or $y$. You can also use the first letter of the unknown value, for instance.
Just look at the equation with the corresponding words.
$\le$ or $\ge$ are used for inequalities.Solution
Let's have a look at what we know:
- Cello has $15$ steaks in total.
- The number of coyotes is unknown. Let's use $c$ to represent the number of coyotes.
- Because each coyote devours one steak, we have to subtract $c$ from $15$, $15-c$.
- $5$ steaks need to be left for the campers to eat. So we must have that $15-c$ equals $5$ in the end.
Find the words which can be represented by variables.Hints
Keep in mind that a variable represents an unknown quantity, which would like to be known.
Perhaps it is helpful to establish the equations which represent each word problem. For example, the first one is given by $(15)(x)=60$.Solution
When solving a word problem, the first step is determining the unknown values which would like to be known, then assigning a variable to each of these unknown values.
Together with the known values you can establish the corresponding equation.
The unknown value is the number of new members. Each given number can't be a variable at all.
In this problem the amount of flour for each muffin is wanted.
You know the number of eggs per portion as well as the total number of eggs. The number of portions is wanted.
Here we know the total number of collected wood. We also know that Mr. Bianca samples 20 pieces. Cello and Jessica collect the same number, which is unknown.
Establish the equation for the given word problems.Hints
Check the known values for each word problem. There is still one unknown value. This is the variable.
First try to establish the equation on your own. Just have a look at the following example:
Jessica prepares a certain number of scrambled eggs portions. For one portion of scrambled eggs she uses $2$ eggs. In total she uses $30$ eggs. This leads to the equation $(2)(x)=30$.
Equations can be written in different ways. Let's have a look at the following example, $a+3=5$.
This equation can also be written as
- $3+a=5$ using the commutative property,
- $3=5-a$, or
- $a=5-3$ using opposite operations.
In each of the given problems there is a wanted value. Each time the corresponding variable is $x$.
Filling the Bus
Because $52$ people are already sitting on the bus and in total $80$ people can fit on the bus, the number of people needed to fill the bus is given by the equation '$52+x=80$.
How Many Buses
We want to know the number of buses. We assign $x$ to this number, and we get the equation $(40)(x)=80$.
How Many People
We have to add the number of males and females together to get the total number of people, or in other words, $28+52=x$.
Because each of the $28$ people have the same number of apples, i.e. $x$, and the total number is given by $84$, then we get the equation $(28)(x)=84$.
Find the variableHints
A variable is often a letter, like $x$, $y$, or any other letter, and is used to represent an unknown quantity.
$+$ or $-$ are operators, while $=$ is a relation.Solution
You can recognize a variable as it is usually a letter (you can also use symbols like a star or a smiley face, but maybe they aren't always the most convenient to use). Looking at our given equations, we can see that:
- $x$, $y$ or $a$, $b$, $c$ are variables.
- $10$, $5$ are numbers.
- $+$ or $-$ are operators.
- $=$ is a relation.
Write the equation.Hints
Consider the unknown value and assign the variable $x$ to it.
Next collect the known values, i.e. the number of kittens as well as the number of toys for each kitten.
You can establish different equations using the commutative property. For example, $x+4=8$ or $4+x=8$.
You can also write the equation above as $4=8-x$ or $x=8-4$.Solution
Sue likes to know the total number of toys. Let's represent this unknown by $x$.
Because she buys $2$ toys for each of her new kittens, $x$ is given by multiplying the given numbers
Sure you can move the order of multiplication using the commutative property to get
You can also divide the equation above by $2$ or $3$ to get
$2=x\div 3$ or $3=x\div 2$.
However, the solution is always the same $x=6$. Sue, enjoy your toy shopping and your time with the new kittens.